2018/4/14
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Contents
0 .
t
'
7th # ksuc .
1 .
4¥ ' HtTHE a hat # at #¥x .
2 . Euler ¥ta a A¥# Aiz 'l¥s4t¥a .
3 .
PEEMEIE'EF# rmz .
4 .
4¥ HE #a a
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IHI
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.
* =a7¥Eirt Ih¥±± F)
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tkukrt # Fta 1- t.EE#TEeIFFET In .

O .
t
'
7th # knuz

i 7th # sit
¥F⇒sb'ts3#¥h#
a # ¥n±kt9kh# Eat A "¥n2n3ta .
531 M =
s2 I a f¥tt9HlFh E = TM .
(Tangent bundle )
TM
~
# '
* Beta .
I>
y
=
¥µTpM :# '
LIMIT
p 1122 : 7 } hli
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-
4pm
YY
: EIIERIF
( pkaF¥isHvtEp¥l )
←
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(
ninth '¥ X e * an a × :
t.Y.IT#ptas
# HE .
)

"
Etants
"
n' 9th #
E¥4¥a '
EMIT E = Mx
Rr
it €53hMknt9Hv#n .
= th 2 As #± A t
'
9th # z Eta # At 9th
'
# said .
^3Ni'
.
4¥ , , A¥# Egg ;qµ*z .
' trivial Vector bundle )
By M =
s
'
In ELKE # .
FZTHTEAKH F¥e*U×R' at 's .
FT IT # RA '
# rtn9HlFkath¥kE£h3 .
RR
-
R
d
s
' -
•
•
•
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4) E = S
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open
Mobius band
C trivial ) ( non . Trivial )

2) to Etna Itau tea IEIAA .
i ) El E = s
'
x IR tiztnktt
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.
3.
'
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ha ( to ) { 53
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qp¥.
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tslikk TELEFE
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95k¥ HE M a
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#'s
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a .
Zn F¥ Fk K Ls ERk2a3 .
4¥ K TM A
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IF ,
M 2 'E#i3 It E¥#a9 .
931 .
s
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a F¥ # a # a A .
*
•
•
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Fit 'U
•
~
•
=
•
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•
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TS
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5×121
'

931 .
5 a t¥ # T5 is Etna itsu .
→ →
tl #aAts5 ,
5 Ii non -
zero then 7khE¥ b"
h¥k3 rtd .
→ Is
→
I→I ( b 'l
'
# 1£ zfnbt i Etsu ! ( taei 'EEAAF3 )
→
€ a 4- It fi -
N KH AT
'
7ft (£EHBHA=) b "
ZEZLF 5 .
a # in 'a#⇒
-15¥ 5×1122

Kd .
K - 52 T
'
a ¥¥ # It Etaa .
T2 = s
'
× s
'
,
TT2= Ts
'
×Ts
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= -12×(112×112) F) ,
¥,
¥
% "
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574£ EIEHZFFHL 5123 .
T -12=5×1122

( £4 e :X na £¥X )
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{ FR 's than ¥¥¥ .
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A #¥ His Hv In a H hit
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it 'E±¥ )
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1 .
4¥44 * A ah #¥ as tE±¥
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HE # A f
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Tsai
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43$33 s ( II ( 2 k£ £ .

HE # East .
3. THE # M tansth # t : E ¥ M KKHZ .
Mn ITHEIII -
¥¥ a "¥tEi CCE ) e HIM ; @ ) { t.tk#2FhF5az
"
.
Az#± to A7Hv # lift hit Ali h#Lz It 'Fi+t3ta .
input output
E
| µ
B
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°
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n'
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7th # .
IKEA Ii Fs¥a¥t¥a .

( "FEFE ) KERI - .
ITHEDI -
rmz
FYFE # M a Th Eat
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Ma IKE K
£H3A#
Eta th 'm { 5-23 A' ¥2 at # FE .
BI .
Hals
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|
.
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|
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0 ( k = 1 ) g
( k=l )
.
( k = 2) ;
( k= 2)
,
-
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( "FEFE ) Thea I - .
IF # a I -
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t
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4¥ # ¥57 sit ( # HE )
Fy THE # M Ian "9th # E httlz .
Mn ITHEDI -
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input output
f
Ka'
tkn
E
( µ
"3
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t.IE#iazY
( Helen )
i 7th # .
ITHERI
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-
¥¥a¥t¥a .

1) Euler #a
FYFE # Mta A±HH5hEi7Hv # t : E ¥ M lift .
E a Euler ¥57 It ,
ECE ) e
HTM
; 7L )
zi .
4nF {
IEAEFEN
.
D e ( ft E) =
fte (E) I AYE 't )
2 e ( EDF ) =
ec E) .
e (F )
3 E : A- a A ⇒ EC E) =
0
4 E : E ATEGTE ⇒ ec E) = -
ECE ) .

2)
Stiefel -
Whitney #A
FYFE # Mta i7Hv # t : E ¥ M KFH ,
E a Stiefel -
Whitney # A { Witt
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.
H .
WICE ) E H
:(
M ; 22 ) ( i. 0,1 ,
...
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)
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1 .
1
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=W(E) -
WCF) ( w=1 + wit . - .
+ Was )
3
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,(8 ,
"
) = L ( Le HYRPIZ. ) =2z
,
4=AiE )

3)
Chern #I
FYFE # Mta
kFE¥ai7Hv
# t : E #M kttl .
E a Chern ¥52 {
Cite
)
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H .
CICE
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...
,r
)
zi .
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IEAEFEN.
@ Co (E) = 1 .
D
Cicft E) =ftc:(E) IEMIIHE)
2
c(EDF ) =
CCE) .
CCF) ( c=nta+ ... +
cn )
3
c.( 8 ,
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d ( de H4 # 2) a 2 ,
EIAIE )

l 7¥¥uEgttkR¥ .
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{
tfk±t2B¥I
:
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Sets
a a
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( ETYIIHI )

2 .
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* .
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.

Euler #aa#I¥Ai .
Rr
FFEHHSKK A 9th # TL :
E M K # L ,
Eo :=
E -
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SEE
,
Mta # Eva
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-
Annett -47*4 ( section ) 2n3 .
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TC
E)
TIE
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= Thom #¥
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HTM )e a
1 ECE ) fruits .
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the# 1 E bl
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ECE ) = 0 .
4¥ 's E A
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Et a A Fsstt
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0 .
i
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zero section
{(2
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cohomology ex ,
seq
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LT ,
des HE ' °
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HTEEO ) HTE ) HMEO )
A
s* s*
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a
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0
B'

# I # 2 77 FE # Ma Euler #x { XCM) 5h32 .
←
Ma¥¥¥a
(
ECTM
) ,
[ M ] > =
XCM ) (
fuetlu
) =
XCM ) )
^Mn¥*
# a Euler #A
MnEler¥I7 EEEAA RB )
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XCM ) =
Io c- isi rank CHICM ) )
i 'a¥x±k3¥¥#k .
4¥ K M A
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= # FIB '¥'t HE F¥ # At a ±n¥h± .
(M ) = V -
E + F .
X
aa¥nh##* ) Katana ) M#ah#¥D

tz .
KTFIHF ( a # At ) if 8 a I # FK Bas't s 5-23 .
•
XC s
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6+4 =
2 .
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0 s if 1274¥ An .
tkbtr theft 1 sy Ts2 If FIAA itsu
!
-
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0 g) ,
t*hL ECTT
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0 { ta2n3 .
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( Fun { Hina Eatx )
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3. PEE '¥FE¥÷t¥ i= suz

per
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it
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( o )
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at 0 not ,
Eitik s z FEBEAFZ PEETE ( obstruction ) 4
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# EE in F '¥#x c : GCM ) → 2 IF 2- cochair 2ta3 .
±5k
Sc(D3
) =
cc
D8
) =
0 ztay ,
2-
cocyde 2ta3 .
i.
€ . as
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c it s NEK '
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-
#k¥F# the Hit 5kt a 9th # a : E # M re HL ,
E n non -
zero section s : M → Eo It
M a v. 1
t.hn#a=tiaEeFhh4i=FFtathtiiI3
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stir
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it 3M£ a¥s 1 PEE '¥¥k
[ c ] e HRCM ; 2)
k£2 K£3 .
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Is : Mcr ,
→ Eo [ c ] =
0
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A
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Euler #A a Ih#
HE ## A -±hfH5hk n' 7th # t : E
*
M KJHL .
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ECE ) e HRCM ; 2)
IF ,
E a non .
zero section E hF3Eh a
* np¥a'¥¥A e -
Exa3 !
CEIIAAIIE )
= KK £ , 2 Euler ¥52 { hE¥x at En .
f3{ . ( E :#a A ⇒ ECE ) .
.
0 ] if ha¥xb 's AA
5101473.

EL F¥k
r=dimM
at¥h¥ .
s : M → Eo e (E) =0
'
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.
:) Ma ✓ Manta it M { atafib 's . B
[ TI En # kktitvkt Xtistutsti 'E3 .
: ) rank ( TM ) = dim M F) ,
In # HA
'
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I 'Is .
XCS
'
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5 I KIF non -
zero vector field ttzktsu . DK

Stiefel .
Whitney #ZaIh#
a-
EL
FEEE ninth # E * M ⇐ HL ,
f , T
•
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→
Ea k -
FFKYE VKCE ) E it E In k ] a
!
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* 1p¥IH÷**k ( mod 2) s -
ZQFZ .

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.
:) W , (E) =
0 # M
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it , (E) = VRCE ) a
section lot 2123 .
c ⇒ M
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A 7th Tan B¥A # Part

¥,¥¥e
Characteristic Classes
Milnor, Stasheff
微分位相幾何学
田村一郎
Lecture Notes in Algebraic Topology
Davis, Kirk
available online!

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@ Take To 9024 .